TY - JOUR

T1 - On problems without polynomial kernels

AU - Bodlaender, Hans L.

AU - Downey, Rodney G.

AU - Fellows, Michael R.

AU - Hermelin, Danny

N1 - Funding Information:
E-mail addresses: hansb@cs.uu.nl (H.L. Bodlaender), rod.downey@vuw.ac.nz (R.G. Downey), michael.fellows@newcastle.edu.au (M.R. Fellows), danny@cri.haifa.ac.il (D. Hermelin). 1 Research supported by the Marsden Fund of New Zealand. 2 Research supported by the Australian Research Council Center of Excellence in Bioinformatics. 3 Supported by the Adams Fellowship of the Israel Academy of Sciences and Humanities.

PY - 2009/1/1

Y1 - 2009/1/1

N2 - Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.

AB - Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.

KW - Composition algorithm

KW - Distillation algorithm

KW - Kernelization

KW - Lower bounds

KW - Parameterized complexity

KW - Polynomial hierarchy

KW - Polynomial kernels

UR - http://www.scopus.com/inward/record.url?scp=70449716028&partnerID=8YFLogxK

U2 - 10.1016/j.jcss.2009.04.001

DO - 10.1016/j.jcss.2009.04.001

M3 - Article

AN - SCOPUS:70449716028

VL - 75

SP - 423

EP - 434

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 8

ER -